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Conjectures of Twin Primes

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Conjectures of Twin Primes By: Christina Mantas Edited and Extended by: Nicole Gillum A prime number is a positive integer that has exactly two factors. • So 1 is not prime, 2 is, and 3 is and 4 isn‟t and so fourth • 2 is the smallest possible difference between two primes The question of whether or not there are infinitely many twin primes is one of the most elusive and longest-standing open questions in number theory. A Twin Prime is a pair of primes that differ by two. • Except for the pair (2, 3), 2 is the smallest possible difference between two primes. • Examples of Twin Primes are: (3,5),(5,7),(11,13),(17,19),(29,31),…,(821,823) In 1849, Alphonse de Polignac, a French mathematician formulated what is know as Polignac‟s Conjecture • Polignac‟s Conjecture states, “For every natural number k, there are infinitely many prime pairs p and q such that p-q=2k." • “There are infinitely many „prime gaps‟ of size 2k” The „prime gaps‟ meaning the difference between two successive prime numbers. The Twin Prime Conjecture is the “k=1” case of Polignac‟s Conjecture It states, “There are infinitely many primes p such that p+2 is also prime” The difference between two primes is called the “prime gap” The first 30 prime gaps are: [1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4, 6,8,4,2,4,2,4,14] • These represent the difference between (2, 3), (3, 5), (5, 7), and so forth. Pairs of primes with prime gap = 4 are called “Cousin Primes” • Examples: (3, 7), (7, 11), (13, 17), and so on Pairs of primes with prime gap = 6 are called “Sexy Primes” • Examples: (5, 11), (7, 13), (11, 17) and so on The last largest known Twin Prime was found January 15, 2007 by Eric Vautier, a French mathematician who submitted his solution to the two distributed computing projects, The Twin Prime Search and Prime Grid. The number is: • 2003663613* 2 to the power of 195000 +/- 1 • The numbers have 58711 decimal digits The current largest known Twin Prime was found on August 6, 2009 by a team of two computer programs and two researchers. It is: • 65516468355·2333333 +/- 1 • The numbers have 100,355 decimal digits The largest known Cousin Primes were discovered in May of 2009 by Ken Davis • p = (311778476·587502·9001#·(587502·9001#+1)+2 10)·(587502·9001#−1)/35+1 where 9001# is called a “primorial” • This number has 11594 digits “9001#” is referred to as “9001 primorial” A primorial works like a factorial but with the composite factors removed from the product. The primorial is the product of only the prime numbers leading up to the number. Example, 9001# = 2*3*5*7*11*13*17*…*9001 Notably, Ken Davis also discovered the largest known Sexy Prime in May, 2009 • p = (117924851×587502×9001#×(587502×9001#+1) +210)×(587502×9001#−1)/35+5 • it has 11593 digits.
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